## An Introduction to Algebraic TopologyThere is a canard that every textbook of algebraic topology either ends with the definition of the Klein bottle or is a personal communication to J. H. C. Whitehead. Of course, this is false, as a glance at the books of Hilton and Wylie, Maunder, Munkres, and Schubert reveals. Still, the canard does reflect some truth. Too often one finds too much generality and too little attention to details. There are two types of obstacle for the student learning algebraic topology. The first is the formidable array of new techniques (e. g. , most students know very little homological algebra); the second obstacle is that the basic defini tions have been so abstracted that their geometric or analytic origins have been obscured. I have tried to overcome these barriers. In the first instance, new definitions are introduced only when needed (e. g. , homology with coeffi cients and cohomology are deferred until after the Eilenberg-Steenrod axioms have been verified for the three homology theories we treat-singular, sim plicial, and cellular). Moreover, many exercises are given to help the reader assimilate material. In the second instance, important definitions are often accompanied by an informal discussion describing their origins (e. g. , winding numbers are discussed before computing 1tl (Sl), Green's theorem occurs before defining homology, and differential forms appear before introducing cohomology). We assume that the reader has had a first course in point-set topology, but we do discuss quotient spaces, path connectedness, and function spaces. |

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### Contents

CHAPTER 0 | 1 |

Brouwer Fixed Point Theorem | 2 |

Categories and Functors | 6 |

CHAPTER 1 | 14 |

Convexity Contractibility and Cones | 18 |

Paths and Path Connectedness | 24 |

CHAPTER 2 | 31 |

Affine Maps | 38 |

Homology and Attaching Cells | 189 |

CW Complexes | 196 |

Cellular Homology | 212 |

EilenbergSteenrod Axioms | 230 |

Chain Equivalences | 233 |

Acyclic Models | 237 |

Lefschetz Fixed Point Theorem | 247 |

Tensor Products | 253 |

CHAPTER 3 | 39 |

The Functor n₁ | 44 |

n₁Są | 50 |

Free Abelian Groups | 59 |

The Singular Complex and Homology Functors | 62 |

Dimension Axiom and Compact Supports | 68 |

The Homotopy Axiom | 72 |

The Hurewicz Theorem | 80 |

Exact Homology Sequences | 93 |

Reduced Homology | 102 |

Homology of Spheres and Some Applications | 109 |

Barycentric Subdivision and the Proof of Excision | 111 |

More Applications to Euclidean Space | 119 |

Simplicial Approximation | 136 |

Abstract Simplicial Complexes | 140 |

Simplicial Homology | 142 |

Comparison with Singular Homology | 147 |

Calculations | 155 |

Fundamental Groups of Polyhedra | 164 |

The Seifertvan Kampen Theorem | 173 |

Attaching Cells | 184 |

Universal Coefficients | 256 |

EilenbergZilber Theorem and the Kunneth Formula | 265 |

Basic Properties | 273 |

Covering Transformations | 284 |

Existence | 295 |

Orbit Spaces | 306 |

Group Objects and Cogroup Objects | 314 |

Loop Space and Suspension | 323 |

Homotopy Groups | 334 |

Exact Sequences | 344 |

Fibrations | 355 |

A Glimpse Ahead | 368 |

Cohomology Groups | 377 |

Universal Coefficients Theorems for Cohomology | 383 |

Cohomology Rings | 390 |

Computations and Applications | 402 |

419 | |

Notation | 423 |

425 | |

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### Common terms and phrases

acyclic algebra Assume axiom basepoint called chain complex chain equivalence chain map closed path coefficients cohomology commutative diagram continuous map convex Corollary CW complex CW decomposition CW subcomplex defined Definition deformation retract denoted disjoint edge path Example Exercise exists fiber fibration finite following diagram commute free abelian group function functor fundamental group gives graded ring group G hence Hint Hom(X homology groups homomorphism homotopy type hTop identified identity inclusion induced injective isomorphism Lemma Let F locally path connected moreover morphism n-simplex natural chain natural map ni(X nullhomotopic open neighborhood open set pair path components pointed map pointed space polyhedron proof of Theorem prove quotient space rank S+(X short exact sequence simplex simplicial map simply connected subgroup subspace surjective topological space triangulation unique universal covering space Vert(K vertex vertices weak topology zero

### References to this book

Computational Homology Tomasz Kaczynski,Konstantin Mischaikow,Marian Mrozek No preview available - 2004 |